[Upitor]

These are fine intros, but then you have to actually dive into these topics. Sometimes there is no real interesting way to explain these topics. Fx, iirc the construction of the real numbers is rather tedious. But I agree, that more effort could be done to motivate many of these topics (at least that was my experience studying math)

[graycat]

"Sometimes"? Yup!

"Tedious"? Yup! Can use Dedekind cuts or maybe something called the normal completion, and especially the first is darned tedious!

There is a good math writer G. F. Simmons, of Introduction to Topology and Modern Analysis, who stated that the two pillars of analysis were linearity and continuity -- nice remark. He also stated that really to understand, have to chew on all the arguments, etc. or some such.

Then I decided to study the proofs really carefully, so to "chew", and in hopes of finding techniques I could use elsewhere. When I mentioned that study technique, objective to my department Chair his remark was "There is no time." -- he also had a point.

Commonly there is an intuitive explanation of what is going on and some views that can provide motivation to study the stuff at all.

There are a lot of books and papers. As a student, I saw a lot of the books, got copies of some of them, put them on my TODO reading list, etc. Eventually, after falling far enough behind on the list, I wondered just where all those books were coming from? It dawned on me, profs need to publish so they do. They are also supposed to have grad students and do. Then the grad students take the advanced course by their major prof and end up with a big pile of notes. Then the grad student, as an assistant prof, wants to publish so cleans up the pile of notes and contacts the usual publishers to publish a book. The top university libraries are essentially required to buy the books, so they get published and bought. And, then, often, there the books sit, gathering dust. I won't say that writing those books was a total waste, and I won't say that students should spend more time reading those books. Or, the books are there on the shelves. They are not really difficult to find. The books have work that was done. Maybe the work is useful now; maybe someday it will be useful; whatever, the work is done, the results found, and there in case they do become useful.

In the meanwhile, back to the mainline of math education, research, applications, usually there can be some helpful intuitive explanations and motivating example applications!

Apparently some authors just give up and assume that their books will mostly just gather dust. But once I wrote Paul Halmos, likely my favorite author, and got back a nice letter from him with "It warms the heart of an author actually to be read, and clearly understood, by ordinary humans." -- at the time I had no academic affiliation and was just reading his book on my own. So, Halmos was surprised that an ordinary human would be reading and understanding his book.

Ah, in what I wrote, I left out that also in linear algebra in n dimensional space, the Pythagorean theorem still holds, that is, an n dimensional version holds!